Fix a language L with two oneplace predicate symbols A and B

Fix a language, L, with two, one-place predicate symbols A and B. For each of the following formulas, find a (simple!) interpretation that makes it true and one that makes it false. Justify your answers. Forall x(A(x) B(x)) Forall x A(x) Forall x B(x) x(A(x) B(x)) x A(x) x B(x) Do the same for Lambda in place of

Solution

a) The statements simply read:

i) For every x, (A(x) is true) or (B(x) is true)

ii) (For every x, A(x) is true) or (For every x, B(x) is true)

iii) There is an x such that (A(x) is true) or (B(x) is true)

iv) (There is an x such that A(x) is true) or (There is an x such that B(x) is true).

Interpretations:

If any of A(x) or B(x) is true for every such x, then i) is true. But if there is an x for which both A(x) and B(x) are false, then i) is false.

If A(x) is true for all x, or B(x) is true for all x, then ii) is true. But if there is an x for which both A(x) and B(x) are false, then ii) is false.

If there is an x such that at least one of A(x) and B(x) is true, then iii) is true. But if A(x) and B(x) are both false for all x, then iii) is false.

If the collection of outcomes A(x) contains one true value for some x, or the collection of outcomes B(x) contains one true value for some (possibly different) x, then iv) is true. But if A(x) and B(x) are false for all x, iv) is false.